Let us shortly summarize the structure and main results of the paper.
Section  collects some basic facts on Gibbs processes based on
the classical point process approach from [1]} and
[2]}. The state space is assumed to be Borel but is
otherwise not required to have any topological properties. The main
result of Section  is the thinning representation
from Theorem REF  which generalizes a result in
[3]}. Our proof is based on Lebesgue–Stieltjes calculus and
is different from the one given in [3]}. In
Section  we prove with Theorem REF  a
version of the thinning representation based on embedding into a
Poisson process on a suitable product space. This result is quite
convenient for coupling purposes and should be compared with the
classical Poisson embedding of marked point processes based on
stochastic intensities; see [5]}. This embedding does not
require any boundedness assumptions on the Papangelou intensity. In
Section  we formulate and study the disagreement
coupling of two (local) Gibbs processes with the same Papangelou
intensity but different boundary conditions. To this end we consider a
symmetric and measurable relation on the state space equipping each
point configuration with a graph structure. At each point of the
state space the Papangelou intensity is then assumed to depend only on
the cluster connected to this point.
Theorem REF  generalizes Theorem 3.1 in
[3]} and shows that the points of disagreement of two
recursively defined point processes are connected to the boundary
conditions. The main part of the proof (which seems to be missing in
[3]}) is devoted to checking that the
coupled processes have the desired Gibbs distribution.
Our main tool here is a basically well-known spatial Markov
property of the Poisson process stopping sets. We work here
with a rather general definition of a stopping
see given in the Appendix of [8]}.
In Section  we discuss empty space probabilities
of Gibbs processes. In particular we show that a large class
of Gibbs processes are Poisson-like as defined in [9]}.
Section  contains some (basically well-known) material on Palm distributions of
a Gibbs process which is needed later. In Theorem REF  we
prove a bound on the total variation distance between an appropriately
scaled thinning of a marked Gibbs process in \(\mathbb {R}^d\)  with
bounded Papangelou intensity and a Poisson process. Our proof adapts a
coupling technique for Poisson approximation from
[10]} to Gibbs processes.
In doing so it exploits the
disagreement coupling studied in Section  for a
Gibbs process and its Palm version.
Similarly as in [11]}, [3]} we assume that
a random graph defined on the points of a dominating Poisson
process does not percolate. In fact we need an exponentially
small cluster size; see (REF ) and (REF )
This condition appears to be weaker than subcriticality
of an associated spatial branching type process, as assumed
in [9]}; see [14]} for a short discussion.
Our general bound involves expectations with respect to
the Gibbs distribution. This cannot be avoided.
Still, apart from subcriticality of the dominating percolation model,
our total variation bound is more explicit and more general than the one
presented in [15]}.
We demonstrate this in Section  by applying
Theorem REF  to Matérn
type I thinnings of Gibbs processes. In this situation
the bound from Theorem REF  becomes very
concrete.
